Английская Википедия:Dini criterion

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Шаблон:Distinguish In mathematics, Dini's criterion is a condition for the pointwise convergence of Fourier series, introduced by Шаблон:Harvs.

Statement

Dini's criterion states that if a periodic function Шаблон:Math has the property that <math>(f(t)+f(-t))/t</math> is locally integrable near Шаблон:Math, then the Fourier series of Шаблон:Math converges to 0 at <math>t=0</math>.

Dini's criterion is in some sense as strong as possible: if Шаблон:Math is a positive continuous function such that Шаблон:Math is not locally integrable near Шаблон:Math, there is a continuous function Шаблон:Math with |Шаблон:Math| ≤ Шаблон:Math whose Fourier series does not converge at Шаблон:Math.

References